=arccos{  Tr[0(t)] },

where we have written \(\rho _0=|\psi _0\rangle \langle \psi _0|\)  and \(\rho (t)=|\phi (t)\rangle \langle \phi (t)|\) .
\(\mathcal {L}(t)\)  is known as the shortest path between two states in the Hilbert space [1]}.
Second, by using (REF ), the dynamics of \(\rho (t)\)  is introduced as follows:


d(t)=d|(t)(t)|+|(t)d(t)|+d|(t)d(t)|
